OP, my living room is very similar to yours.
We recently did ceiling height built in bookshelves either side of the fire place with log and toy storage underneath as you suggest. I am really pleased with it.
We have floated our sofa like you but we put in a simple end table with a shelf at the door end of the sofa. In front of the sofa we have a small coffee table. My dad made it from an Ana White woodworking plan we found online. It has a shelf underneath with baskets but best part is the hidden storage under the table top that hold board games and all manner of junk.
On the wall behind the sofa we put in two IKEA Hemnes shoe cabinets with a large mirror above. The mirror makes the room feel bigger and with the addition of a few coat hooks behind the door and a runner we have managed to create a clutter-free mini hallway.
IKEA has a lot of really slim storage solutions that could work in that space behind your sofa. We have the pull out shoe cabinet and use it for everything from shoes, hats and gloves, sunscreen etc, to chargers, cables and all those little bits and pieces that need a home. They also have slim cabinets with flat shelves that can be stack or go side by side to maximise storage. Check out their Besta, Mackapar and Eket lines, they are all really clean and simple and don't scream IKEA like the ubiquitous Kallax.
I have found that in such a small space have doors on cabinets and bookshelves and baskets on open shelving really helps declutter.
My house is tiny and it can be really challenging to make it work for everyday family life but IKEA is a godsend and if you chose wisely, it can look great.